Results 1 to 4 of 4

Math Help - show the expressions are rational numbers

  1. #1
    Senior Member
    Joined
    Jan 2007
    Posts
    476

    show the expressions are rational numbers

    give a, b and c are rational numbers, show the expressions are rational.

    i.) a^2 + b^2 + c ^2

    ii.) (5a^3 +6b^4)/(a^2+b^2)

    iii.) a + a^2 + a^3 + ... +a^10

    I had alot of trouble proving this, and it was not enough work.

    please explain these.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by rcmango View Post
    give a, b and c are rational numbers, show the expressions are rational.

    i.) a^2 + b^2 + c ^2

    ii.) (5a^3 +6b^4)/(a^2+b^2)

    iii.) a + a^2 + a^3 + ... +a^10

    I had alot of trouble proving this, and it was not enough work.

    please explain these.
    Write a, b and c as ratio of integers, then substitute these into the expressions and rearrange to show that they are the ratios of integers.

    For example let a=e/f, b=g/h, c=i/j, where e,\ f,\ g,\ i and j are integers.

    Then:

    a^2 + b^2 + c ^2=\frac{e^2}{f^2}+\frac{g^2}{h^2}+\frac{i^2}{j^2}  =\frac{e^2h^2j^2+g^2f^2j^2+i^2f^2h^2}{f^2h^2j^2}

    RonL
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Mar 2008
    Posts
    148
    Speaking generally there are a lot of different operations one can do to rational numbers without leaving the rational world.

    The product of any finite number of rational numbers is rational:
    let  q_1, q_2, ... q_k be rational numbers.
    say  q_i=\frac{n_i}{m_i},  \forall i, \text{ and } m_i, n_i are integers.
    then n_1n_2...n_k \text{ and } m_1m_2...m_k are integers
    therefore q_1 q_2 ... q_k is rational.

    Same kind of proofs can be used to show the sum of 2 rationals is rational, which extends inductively to the finite sum of rationals. The division of rationals remaining rational is a direct consequence of the multiplication of rationals remaining rational. So it is not hard to see that any finite composition of such operations on rationals results in a rational. Proving these general statements might be easier than specific examples which can get very messy.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Jan 2007
    Posts
    476
    iii.) a + a^2 + a^3 + ... +a^10

    i could let a = f/e

    then = (f/e)^2

    then no matter what to the 10th power it is rational.

    is that correct?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Show the set of rational numbers Q is neither open nor closed.
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: February 23rd 2011, 08:18 PM
  2. Show that rational numbers correspond to decimals
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: February 11th 2011, 04:50 AM
  3. Rational Expressions and Radical Expressions
    Posted in the Algebra Forum
    Replies: 16
    Last Post: July 9th 2009, 09:29 PM
  4. Rational Expressions
    Posted in the Algebra Forum
    Replies: 2
    Last Post: March 2nd 2009, 04:42 PM
  5. rational expressions
    Posted in the Algebra Forum
    Replies: 7
    Last Post: December 20th 2006, 11:15 PM

Search Tags


/mathhelpforum @mathhelpforum